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NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Indefinite Integration

NTA Abhyas 2020 · Mathematics · Indefinite Integration

Let A _ n = ∫ ^ n xdx, n ∈ ℕ . If A _ 10 + A _ 12 = ^ m x m + λ...

Let An=∫tan⁡nxdx,∀n∈N{A}_{n} = ∫ \tan^{n} xdx, \forall n \in \mathbb{N}. If A10+A12=tan⁡mxm+λ{A}_{10} + {A}_{12} = \dfrac{\tan^{m} x}{m} + \lambda (where λ\lambda is an arbitrary constant), then the value of mm& is equal to

Options

  1. A1010
  2. B1111
  3. C1212
  4. D1313

Correct answer

B. 1111

Step-by-step solution

A10+A12=∫tan⁡10xdx+∫tan⁡12xdx{A}_{10} + {A}_{12} = ∫ \tan^{10} xdx + ∫ \tan^{12} xdx
=∫(tan⁡10x+tan⁡12x)dx= ∫ \left(\tan^{10} x + \tan^{12} x\right)dx
=∫tan⁡10x(1+tan⁡2x)dx= ∫ \tan^{10} x\left(1 + \tan^{2} x\right)dx
=∫tan⁡10x⋅sec⁡2xdx= ∫ \tan^{10} x ⋅ \sec^{2} xdx
Let, tanx=t⇒sec⁡2xdx=dttan x = t ⇒ \sec^{2} xdx = dt
∴A10+A12=∫t10dt=t1111+λ∴ {A}_{10} + {A}_{12} = ∫ t^{10}dt = \dfrac{t^{11}}{11} + \lambda
=tan⁡11x+λ11= \dfrac{\tan^{11} x + \lambda}{11}
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